ENGINEERING TRANSACTIONS
No. x
2014
FROM TOLERANCE MODELLING TO EXACT
DESCRIPTION OF HEAT CONDUCTION IN BIPERIODIC
COMPOSITES
EwarystWierzbicki, Marta Mazewska
Warsaw University of Life Sciences – SGGW
Faculty of Civil and Environmental Engineering
In the paper we are to study the temperature boundary effect behavior in the dividing
wall made of the composite conductor in which every area parallel to the outside and the
inside plane areas has identical biperiodic structure. The proposed modeling approach is
restricted to the hexagonal case of bi-periodicity in which the hexagonal cell can be
divided onto three rhombus parts with different thermal properties. Considerations deal
with anisotropic conductors.
Keywords:tolerance averaging, heat conduction, biperiodic conductors,
boundary effect
1. INTRODUCTION
In many problems of environmental engineering we deal with highly
oscillating heat transfer through walls built with different types of materials. An
example of this phenomena is the presence of disturbances of a constant
temperature in the external air boundary layer localized close to walls
constructed from materials with a dense microstructure. Unfortunately, to
compare the conductor responses on the different types of oscillating boundary
thermal loads, the approximated models are inadequate. That is why the
investigation of exact description of a heat conduction in composite materials is
justified. The tolerance modelling technique explained by prof. Cz. Woźniak
and contributors in [5,12,13,14] may lead to the feasibility of this type of
comparison, provided that the used tolerance shape functions lead to situations
in which the heat flux vector generated by the temperature approximation given
by micro-macro hypothesis retain continuity of its component normal to the
2
Ewaryst WIERZBICKI, Marta MAZEWSKA
bonding surface of composite components. Modeling problems for the heat
conduction and for the linear elasticity in hexagonal-type composites have been
investigated in [1,8,11].
Considerations will be focused on the damping effect of external
temperature fluctuation by the wall built of the mentioned composite
material.Such behavior is usually referred to as the temperature effect behavior.
2. SUBJECT OF CONSIDERATIONS
Let us consider a special type of biperiodic hexagonal type structure in
which every basic hexagonal cell  is divided into three rhombus parts
 1 ,  2 ,  3 , cf. Fig.1. Each of these three rhombus parts has different
properties described below. The hexagonal cell is situated in Carthesian
orthogonal coordinate system Ox 1 x 2 x 3 in which the plane Ox 1 x 2 is a
biperiodicity plane and Ox 3  Oz is normal tobiperiodicity plane.
Thermal properties of the composite is described by a heat conductivity
tensor field represented in the introduced coordinate system by symmetric
matrix
 k11
K  K ( x)  k 21
k 31
k12
k 22
k 32
k13 
k 23 ,   [k13 , k 23 ]T
k 33 
(2.1)
taking constant values K I , K II , K III in rhombus  1 ,  2 ,  3 , respectively,
and by the scalar field c () of specific heat taking related constant values c I , c II ,
c III in rhombus  1 ,  2 ,  3 .
Fig. 1. Basic hexagonal cell
ON THE BOUNDARY EFFECT BEHAVIOR IN THE HEXAGONAL-TYPE
BIPERIODIC DIVIDING WALLS
3
33
We shall assume jump discontinuitiesof K ()  R
and c I , c II , c III on
interfaces between every neighboringrhombus parts as well as continuity of the
heat flux vector in directions normal to the planes separating components.
Hence,composite components are perfectly bonded [9,10].
This work presents two special cases of considered hexagonal-type
structure. In the first case we introduce three rhombus parts with isotropic
properties (Fig.2.) marked by heat conductivity tensor in form:
1 0 0
K a  k a I  k a 0 1 0, a  I , II , III
0 0 1
(2.2)
Fig. 2. Basic hexagonal cell with three isotropic rhombus parts
In the second case the heat conductivity tensor of the first rhombus part of the
basic hexagon will be assumed in the form

 sI
 I
k
K 




1

sII
k II
0
0
sI k I  sII k II
0
0





0
I
II 
sI k  sII k 


0
in which s1  s2  1 , s1 , s2  0 and k I , k II are two conductivity constants of
some isotropic conductors.
(2.3)
4
Ewaryst WIERZBICKI, Marta MAZEWSKA
The subsequent two rhombuses, second and third, are assumed to be made
of the same transversally anisotropic conductor determined by conductivity
matrix (1.3) but rotated in every neighboring rhombus by an angle of 2π/3
(Fig.3.) with 0z axis as the axis of rotation. Hence
 K for ( x 1 , x 2 )   I

T
K a  Q2 / 3 K Q2 / 3 for ( x 1 , x 2 )   II

T
1
2
Q2 / 3 K Q2 / 3 for ( x , x )   III
(2.4)
Fig. 3. Basic hexagonal cell with rotational symmetry
Symbol Q2 / 3 denotes here the orthogonal matrix of rotation by 2 / 3 in R 2 ,
namely
cos 
Q   sin 
 0
 sin 
cos 
0
 1

2  2

0  
3  3
0   
 2
1 
 0


3
2
1

2
0


0


0

1


(2.5)
Material, which conductivity properties are described by conductivity matrix
(2.3), can be interpreted as two-phased periodic layer with dense periodic
structure. Under this interpretation s1  l1 / (l1  l2 ) , s2  l2 / (l1  l2 ) , where
l1 , l2 are the thicknesses of the first and the second layer, respectively.
ON THE BOUNDARY EFFECT BEHAVIOR IN THE HEXAGONAL-TYPE
BIPERIODIC DIVIDING WALLS
5
The basic difficulty of the tolerance modeling procedure is in a proper(for
every considered physical problem) choice of the shape function system. This
problem will be investigated in the subsequentsections.
3. MODELLING PROCEDURE
The starting point of the tolerance modeling of heat conduction in periodic
composites is the idea of the description of the temperature field by the finite
slowly-varying fields u, A , A  1,..., N , and introducing the micro-macro
approximation of the temperature field
   M  u   g A A
(3.1)
into a parabolic heat transfer equation
c  (  )  [ K (  )]  b
(3.2)
In (3.1) and wherever we deal with repeated indices summation convention
holds. New introduced above fields, averaged temperature field u and
fluctuation amplitude fields  A are new basic unknowns in the tolerance model
equations. In (2.1) symbol b means the heat sources field and θ is temperature
  grad   for
field. There is also denoted
grad  [1 ,  2 ,0]T and
1
  [0,0,  3 ]T where  1   / x ,  2   / x 2 ,  3   / x 3 . The tolerance
modeling procedure yields to the model equations
c  (   )  [ K  u   K grad g A  A ]  b
 2 ( g Acg B  B  T  g A Kg B  B ) 
(3.3)
 ( g grad g K    g grad g K  ) B 
A
T
B
B
T
A
 grad T g A K grad g B  B   grad T g A K u    g Ab
in which the standard averaging  ( x ) over cells x   has been applied. For
particulars the reader is referred to [6,7,12]. Fluctuation amplitudes in tolerance
model equations (2.3) describe the suppressions of the temperature perturbations
togetherwith the growthofthe distance from the boundary of the region occupied
by the composite.
In the subsequent considerations an important role plays the heat flux
6
Ewaryst WIERZBICKI, Marta MAZEWSKA
qM  K (u   g A A )
(3.4)
generated by the micro-macro approximation (3.1) of the temperature field. This
special kind of heat flux is referred to as the tolerance heat flux vector. If the
component of the tolerance heat flux vector, normal to the surfaces 
separatingingredients of the composite, is continuous on these surfacesthen
wesay thatthe tolerance heat flux continuity condition is satisfied.
4. BOUNDARY EFFECT EQUATION
The theory of linear ordinary differential equations suggests to investigate the
fluctuation amplitude  A in the form of the decomposition  A   A  A onto
two terms  A and  A . The first part  A should satisfy equation
 2 ( g Acg B  B  T  g A Kg B  B ) 
 ( g A grad T g B K    g B grad T g A K  ) B 
(4.1)
 grad T g A K grad g B  B  0
while the second part should be an arbitrary solutions to
 2 ( g Acg B  B  T  g A Kg B  B ) 
 ( g A grad T g B K    g B grad T g A K  ) B 
(4.2)
 grad T g A K grad g B  B   grad T g A K u    g Ab
Equation (4.1) will be referred to as the nonstationary boundary effect equation.
At the same time equation
 2 T  g A Kg B  B ) 
 ( g A grad T g B K    g B grad T g A K  ) B 
(4.3)
 grad T g A K grad g B  B  0
will be referred to as the stationary boundary effect equation or simply the
boundary effect equation, cf. [2,3,4].
5. SHAPE FUNCTIONS
ON THE BOUNDARY EFFECT BEHAVIOR IN THE HEXAGONAL-TYPE
BIPERIODIC DIVIDING WALLS
7
In the tolerance modeling technique a proper indication of the finite sequence
h1 ( x), h2 ( x),..., h N ( x) of tolerance shape functions is the most important
modeling problem. In accordance with the principles of the tolerance modeling,
this indication is verified only by a posteriori evaluation of the accuracy of the
obtained solutions. In addition to the formal conditions required to be met by a
string tolerance shape functions, there are no known guidelines for constructing
them.
In this paper we are to investigate the tolerance shape functions leading to
the tolerance heat flux continuity condition formulated at the end ofthe previous
section. For the anisotropic conductors the existence of such tolerance shape
functions is still an open mathematical problem. Such shape functions plays a
crucial role in the the extended tolerance model, [7], in the framework of which
the exact description of the heat transfer in composite materials can be obtained.
6. PASSAGE TO THE LIMIT MODEL
Under the assumption that the tolerance heat flux continuity condition for the
tolerance shape functions h1 ( x), h2 ( x),..., h N ( x) is satisfied we shall pass from
the tolerance model represented by (3.3) to the extended tolerance model. Such
procedure has been realized for two-phased periodic laminated composites in
[17].
In the first step of this procedure we shall introduce instead of micromacro hypothesis (3.4) a new hypothesis in the form
(6.1)
 ( x, z, t )  M ( x, z, t )  am,n ( z, t ) m,n ( x) .
for  m,n ( x) 
2
4
 m ( x ) n ( x ) and for
 m ( x )  cos
2n
x ,
3
x  0,
3
,
2
 n ( x )  cos
2n
x ,
3
x  0,
3
, .
2
(6.2)
1
3
x2 
( x1  1)
2
2
where in long wave term  M  u  h A ( x) A we shall assume that fluctuation
x 
amplitudes  A   A ( z, t ) do not depend on periodic coordinates x  ( x1 , x2 ) and
field u  u ( x, z, t ) is not restricted by the slowly-varying condition, cf. [1]. At
the same time  res     M ( x, z , t ) is, under the tolerance heat flux continuity
8
condition
Ewaryst WIERZBICKI, Marta MAZEWSKA
a
certain
continuously
differentiable
scalar
field
and
res ( x, z, t )  a p ( z, t ) p ( x) , p  (m, n) , is the Fourier expansion with
coefficients am ,n  am ,n ( z , t ) and with respect to the orthogonal basis
 m , n ( x) 
2
m(
x
) n (
x
).
4


In the second step of the mentioned procedure we shall assume that the
orthogonal system of functions  m,n ( x) is independent of the thermal and
geometrical properties of the conductor. Hence, m, n  0 in (6.1) and
  h A  0 and   p  0 almost everywhere on  . That is why
c p   0,  K p   0,
p  1,2,... ,
ch A   0,  Kh A   0,
A  1,..., N .
(6.3)
After introducing new representations h ( x)  g ( x) for tolerance shape
A
A
1
functions h A ( x) and  ( x)   ( x) , p  (m, n) ,
assumptions lead to the following extended model equations
p
p
1
aforementioned
 cu   T [ K u   K   g  A   K   p  a p ]  b
 gAcg B   g Ac p    B 

p
B
p
q 
a


cg



c


q

 
 2 {
 
T
   gAcg B   g Ac p    B  
  p B
 } 
p
q 
  Kg   c   aq  
  grad T g A Kg B   grad T g A K p  
  

  grad T  p Kg B   grad T  p k q  


  grad T g B Kg A   grad T  q Kg A     B 


T B
p
T q
p 
 aq 

grad
g
K



grad

K





 grad T g A Kgradg B   grad T g A Kgrad  q    B 


T p
B
T p
q 
  grad  Kgradg   grad  kgrad    aq 
 grad T g A K u  
  g A b 

 p 
T p
 grad  K u  
 b 
(6.4)
ON THE BOUNDARY EFFECT BEHAVIOR IN THE HEXAGONAL-TYPE
BIPERIODIC DIVIDING WALLS
9
A formal procedure of rescalling tolerance shape functionsleads to the limit
model equations
 cu   T [ K u   K   g  A   K   p  a p ]  b
0   B 
0

q 
p
0  c    aq 
 2 {
 0
0   B  
}
 T  
p
 0  c q    aq  



  0  grad T g A K q  

  
 0  grad T  p k q  


0
0
    B 



p 
T q
p
T B
  grad g K   grad  K      aq 
 grad T g A Kgradg B   grad T g A Kgrad  q    B 


q 
T p
B
T p
  grad  Kgradg   grad  kgrad    aq 
  g Ab 
 grad T g A K u  

 p 
T p
 b 
 grad  K u  
(6.5)
The limit model of the heat conduction in the considered isotropic
hexagonal-type composite represented by equations (6.5) is valid provided that
the system of tolerance shape functions h1 , h 2 ,…, h N leads to the tolerance
heat flux continuity condition.
Let rot ( x)  Q ( x  x0 )T  x0 , for x  R 2 ,   R , and for and
arbitrary cell center x0 as a center of rotation. In the subsequent considerations
we shall assume that three shape functions
2
1
3
2
h1 ( x)   g ( 1 x) , h ( x)  h (rot2 /3 ( x)) , h ( x)  h (rot2 /3 ( x)) (6.6)
determined by a single basis function g () is taken as the system of tolerance
shape functions. In [16] the basis function g () satisfying condition (6.6) and
the tolerance heat flux continuity condition . Hence, the limit model of the heat
conduction in the considered anisotropic hexagonal-type composite represented
by equations (6.5) is valid.
7. FINAL REMARKS
10
Ewaryst WIERZBICKI, Marta MAZEWSKA
The answer to the question whether amplitude fluctuations  A can be
eliminated from the limit model equations (6.5) is still not known. It
means that the answer to the question whether the short-term and long-term
temperature boundary disturbances are transmitted through the hexagonal-type
conductor independently is also an open problem. It must be emphasized that for
two-phased laminated periodic conductors this problem has been solved in [17]
and the exact description of the boundary effect phenomena in this case has been
obtained.
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ON THE BOUNDARY EFFECT BEHAVIOR IN THE HEXAGONAL-TYPE
BIPERIODIC DIVIDING WALLS
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